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Moment Distribution Method
Lesson Objectives:
- Identify the formulation and sign conventions associated with the Moment Distribution Method.
- Derive the Moment Distribution Method equations using mechanics and mathematics.
- Outline procedure and compute the structural response via Moment Distribution Method.
Background Reading:
- Read ___________________________________________________________________
Moment Distribution Method Overview:
- This method was first introduced in _______________ by _________________________ for the analysis of ________________________________________________________.
- This is another classical formulation of the ____________________________________.
- This method only considers ________________________________________________. a. Therefore the assumption is made that __________________________________ are negligible. b. Reasonable? _______________________________________________________
- The moment distribution method is useful in the approach to perform structural analysis: a. Gain __________________________ into the structural __________________ and ___________________________. b. Does not require to solve a ___________________________________________. i. This is required within the _____________________________________. c. Useful method to check _____________________________________________.
Moment Distribution Method Basics:
- In order to solve the structure’s system of ___________________________equations simultaneously: a. The _______________________________________ is examined at a single joint. b. This is performed in an ______________________________________________.
- A consistent sign convention is established where: a. _________________________________________________are positive when _________________________________________. b. _________________________________________________are positive when _________________________________________.
- To perform the ______________________________, one needs to identify how the moment distributes at a __________________________________________. a. The term ___________________________________________ is introduced and this is a function of the ________________________________ of members that frame the joints. b. Generally this is written as:
Derivation of Member Stiffness Values:
- Let’s consider two different member fixities, ______________________________ and ______________________________ to derive the member stiffness values.
- Recall from the slope-deflection notes, a ______________________________________ is defined as the moment developed at ________________________________ for an applied external moment ______________________________________. a. For pinned-fixed, the __________________ is:
b. For pinned-hinged, the __________________ is:
- Since ______ is horizontal, one can apply the first moment area theorem as:
a. Relates the applied end moment and rotation of the corresponding end.
Now, lets consider the second type of member. A sketch of the second example member AB where the ends are _________________________:
Curvature diagram sketch of example member AB:
From examination of the elastic curve, the rotation at the near end can be written as:
Assuming the member is prismatic, the second moment area theorem can be applied as:
Now with the two aforementioned relationships noted (equations ____ and ____), let’s summarize the bending stiffness values.
Let ________ be defined as the bending stiffness of a member. a. The ___________________ required at one end of the member to cause a ______ _______________________ at that (same) end).
For a ____________________________________ member, this bending stiffness can be expressed as:
If _________________________________ is constant, a newly defined _____________ ______________________ (denoted as _____) is:
Distribution Factors Introduction and Derivation:
A distribution factor is defined as the ratio of ___________________________________ of each member to the sum of all the __________________________________________ at that joint. a. This is commonly denoted as ____________.
Sketch of a simple three-member frame structure with an externally _________________ _______________________:
The free body diagram (FBD) can be drawn as:
The free body diagram (FBD) of joint B is:
Writing the _____________________ equilibrium equation:
Note that for the three members: a. Member _____ is _________________________________________. b. Member _____ is _________________________________________. c. Member _____ is _________________________________________.
With previous knowledge of the carryover moment, three expressions for end moments can be written as:
- This can be generalized with the definition of a _________________________________. a. A distribution factor is defined as the ___________________________________ applied to the_________________________________ at end B of member i. b. Denoted and written in basic form as:
Fixed-End Moments:
- Just as used in the _________________________________, expressions for fixed-end are also required for the moment distribution method.
- Unlike the _________________________________, the effects due to ______________ _________________________ and ____________________________________ must be accounted for using FEM.
- Example of FEM due to weak foundations/support settlements:
Procedure for Moment Distribution Method:
- Calculate the distribution factors (______). Check that _________________ at each joint.
- Compute the fixed end moments. Recall the sign-convention such that _______________ _______________ FEM are established as positive.
- Balance the moments at each joint that is free to rotate in an iterative approach: a. At each joint: first evaluate the ________________________________ and distribute to each member using the ____________________________________. b. Perform the _______________________________________________________. c. Repeat as required until _____________________________________________.
- Determine the final member end moments by the sum of __________________________ and ________________________ moments. a. Note that moment equilibrium must be satisfied for joints that are _____________ ___________________________.
- Compute the member end shears by __________________________________________.
- Compute the support reactions at joints using _____________________________.
- Check the calculations of the end shears and support reactions using equilibrium.
- Draw the shear and bending moment diagrams, if required.
- Compute Fixed-End Moments.
ܯܧܨ ൌ
- Balance moments at joints and determine final end moments. Begin the moment distribution process by balancing joints B and C. At joint B: ܯ௨ௗ^ ܯܧܨ ൌ ܯܧܨ ൌ
ܯܦ ܨܦ ൌ ܯ൫െ௨ௗ^ ൯ ൌ
ܯܦ ܨܦ ൌ ܯ൫െ௨ௗ^ ൯ ൌ
At joint C:
ܯ௨ௗ^ ܯܧܨ ൌ ܯܧܨ ൌ
ܯܦ ܨܦ ൌ ܯ൫െ௨ௗ^ ൯ ൌ
ܯܦ ܨܦ ൌ ܯ൫െ௨ௗ^ ൯ ൌ
Perform the carryover moments at the far ends of each beam segment. Due to the distributed moments at joint B: ܯܱܥ (^) ൌ^12 ܯܦሺ ሻ ൌ
ܯܱܥ ൌ^12 ܯܦሺ ሻ ൌ
Due to the distributed moments at joint C:
ܯܱܥ (^) ൌ^12 ܯܦሺ ሻ ൌ
ܯܱܥ ൌ^12 ܯܦሺ ሻ ൌ
Now repeat until the unbalanced moments are negligibly small. It is simple to perform this task in a tabular form (reference Table 1):
Special Cases of the Moment Distribution Method:
- Simple support at one end. a. Sketch:
b. Recall that the bending stiffness is:
c. At the simple support, the distribution factor is _____.
d. To apply the moment distribution method, balance the joint only _____________. Leave the free end ___________________, where the moment is ____________.
- Cantilever overhang at one end. a. Sketch:
b. Section _____ does not contribute to the rotation stiffness at joint ________. c. At the free end, the distribution factor is _____. d. However, the loads on the cantilever must be still considered at joint _______.
Sketch of the frame with ___________________________________________________ (___________________________):
Sketch of the frame with ___________________________________________________ (___________________________):
Procedure for Moment Distribution Method for Frames with Sidesway:
First solve the frame with external loads for ___________________________________. a. Find _____________________________.
Find the fictitious reaction (___________________________________) by equilibrium.
Analyze a second frame with the fictitious reaction applied in the opposite direction.
Superimpose the relationship of:
However, one cannot directly compute the values of ______. This requires an in-direct approach.
Analyze a third frame structure with an unknown translation (______________), under an externally applied load of unknown magnitude ______. a. Note _____ is in the opposite direction of ______.
To solve easily, assume that “____” corresponds to an FEM. Find the remaining FEM values and perform the moment distribution method. a. Find _____________________________.
Find the value of load _____ by equilibrium.
The developed moments are linearly proportional to the magnitude of the load. a. Therefore find the ratio of:
Assemble the frame by superimposing the loads with the equation:
